Euler's mathematics in terms of modern theories? Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in operation in their writings a conception of mathematics which is quite extraneous to that of Euler." Ferraro concludes that "the attempt to specify Euler's notions by applying modern concepts is only possible if elements are used which are essentially alien to them, and thus Eulerian mathematics is transformed into something wholly different"; see http://dx.doi.org/10.1016/S0315-0860(03)00030-2. 
Meanwhile, P. Reeder writes: "I aim to reformulate a pair of proofs from [Euler's] "Introductio" using concepts and techniques from Abraham Robinson's celebrated non-standard analysis (NSA). I will specifically examine Euler's proof of the Euler formula and his proof of the divergence of the harmonic series. Both of these results have been proved in subsequent centuries using epsilontic (standard epsilon-delta) arguments. The epsilontic arguments differ significantly from Euler's original proofs." Reeder concludes that "NSA possesses the tools to provide appropriate proxies of the inferential moves found in the Introductio"; see http://philosophy.nd.edu/assets/81379/mwpmw_13.summaries.pdf (page 6). 
Historians and philosophers thus appear to disagree sharply as to the relevance of modern theories to Euler's mathematics. Can one meaningfully reformulate Euler's infinitesimal mathematics in terms of modern theories?
Note 1. There is a related thread at Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite?
Note 2. We challenged a reductionist view of Euler's infinitesimal mathematics in a recent article in The Mathematical Intelligencer.  Here we refute H. Edwards' reduction of Euler to an Archimedean framework.
 A: [Converted from comment to answer per Yemon Choi's suggestion.]
From a casual run-through of the Ferraro paper, it seems like Euler's ideas about infinitesimals were, unsurprisingly, not formalized to modern standards and therefore don't map exactly onto modern concepts. He apparently didn't think of a line segment as a point set, which would be more similar to smooth infinitesimal analysis than to NSA. But other aspects of Ferraro's description do seem more like NSA than SIA. Infinite numbers are imagined as infinitely increasing sequences, whereas not all models of SIA have invertible infinitesimals. I assume Euler used Aristotelian logic. 
A: A somewhat delayed response is provided in our detailed study of Euler accepted for publication in Journal for General Philosophy of Science.
We apply Benacerraf's distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the 17th and 18th century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass's ghost behind some of the received historiography on Euler's infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a "primary point of reference for understanding the eighteenth-century theories." Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler's own.
Euler's use of infinite integers and the associated infinite products is analyzed in the context of his infinite product decomposition for the sine function. Euler's principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. 
We argue that Ferraro's assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler's work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. 
